Sample Size Calculator

Proportion: n = z² · p̂ · (1 − p̂) / E². p̂ is your best guess of the true proportion (use 0.5 for the most conservative — largest required n). E is the half-width of the confidence interval you'll accept.

Mean: n = (z · σ / E)². σ is the population standard deviation (use a pilot study or domain knowledge to estimate). E is the half-width of the interval on the mean.

Finite-population correction (proportions): n_adj = n / (1 + (n − 1)/N) where N is the population size. Drops n meaningfully when n approaches N.

You want to estimate the proportion of voters supporting a candidate within ±3 percentage points, at 95% confidence.

z = 1.96, E = 0.03, p̂ = 0.5 (worst case).

n = 1.96² · 0.5 · 0.5 / 0.03² = 3.8416 · 0.25 / 0.0009 ≈ 1067.

Halve the margin to ±1.5 pp and n quadruples to ≈ 4269. Precision is expensive.

Estimate the mean weight of bagged chips to within ±2 g at 99% confidence. Historical σ = 8 g.

z = 2.576, n = (2.576 · 8 / 2)² = 10.304² ≈ 106.

If σ is unknown, the formula uses z; the actual study then uses t. For n > 30 the difference is small; for smaller pilot runs, inflate by ~10–15% as a safety margin.

Quadrupling confidence width has no fixed cost; doubling precision (halving E) always quadruples n.

Margin of error scales as 1/√n. Going from n = 400 to n = 1600 halves the margin.

If the true p̂ is close to 0 or 1, the required n is smaller than the p̂ = 0.5 worst case — but unless you're confident about p̂, use 0.5 to be safe.

Ignoring non-response. If 60% of contacted respondents reply, you need to contact 1067 / 0.6 ≈ 1779 people to land 1067 responses.

Stratified or clustered designs need a design effect (DEFF) multiplier, often 1.2–2.0, on top of the simple-random-sample n.

Multiple subgroup estimates each need their own n. A national poll with N = 1067 can have ±3pp overall but ±10pp for a 100-person subgroup.

Forgetting the FPC when N is small relative to n. Sampling 400 of 1000 needs only n_adj ≈ 286.